For each linear transformation below, state a basis for Ker(T), the Nullity of the transformation, and the Rank of the transformation. If you do not need a vector, then place zeros for all entries of that vector (for example, if you only need 2 vectors for the basis, then fill in the first two vectors and make all subsequent vectors have 0 for all boxes). a 1) Let T b - B - L la+1b+1c 2a+3b+3c] -la + (-2)b+ (-2)c 3a+3b+3c i) A basis for Ker(T) would be: ii) The Nullity of T is: iii) The Rank of T is: 2) Let T(a + bx + cx² + dx³) = i) A basis for Ker(T) would be: ii) The Nullity of T is: iii) The Rank of T is: 3) Let T a b [ d f [def = 1a2b+1c + 1d [la+16+2c+2d 2a+36 + 4c+ 2d -la + (-3)6 + (-1)c+2d] la+(-1)b+1c+ (-1)d + 2e+4f 2a+(-1)b+4c+4d + 6e + 12f Oa+(-1)b+(-1)c+ (-4)d + (-1)e + (-2) f 6a+(-3)b+11c + 10d + 18e + 34f i) A basis for Ker(T) would be:
For each linear transformation below, state a basis for Ker(T), the Nullity of the transformation, and the Rank of the transformation. If you do not need a vector, then place zeros for all entries of that vector (for example, if you only need 2 vectors for the basis, then fill in the first two vectors and make all subsequent vectors have 0 for all boxes). a 1) Let T b - B - L la+1b+1c 2a+3b+3c] -la + (-2)b+ (-2)c 3a+3b+3c i) A basis for Ker(T) would be: ii) The Nullity of T is: iii) The Rank of T is: 2) Let T(a + bx + cx² + dx³) = i) A basis for Ker(T) would be: ii) The Nullity of T is: iii) The Rank of T is: 3) Let T a b [ d f [def = 1a2b+1c + 1d [la+16+2c+2d 2a+36 + 4c+ 2d -la + (-3)6 + (-1)c+2d] la+(-1)b+1c+ (-1)d + 2e+4f 2a+(-1)b+4c+4d + 6e + 12f Oa+(-1)b+(-1)c+ (-4)d + (-1)e + (-2) f 6a+(-3)b+11c + 10d + 18e + 34f i) A basis for Ker(T) would be:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![For each linear transformation below, state a basis for Ker(T), the Nullity of the transformation, and the Rank of the transformation. If you do not
need a vector, then place zeros for all entries of that vector (for example, if you only need 2 vectors for the basis, then fill in the first
two vectors and make all subsequent vectors have 0 for all boxes).
a
1) Let T b
- B - L
la+1b+1c
2a+3b+3c]
-la + (-2)b+ (-2)c 3a+3b+3c
i) A basis for Ker(T) would be:
ii) The Nullity of T is:
iii) The Rank of T is:
2) Let T(a + bx + cx² + dx³) =
i) A basis for Ker(T) would be:
ii) The Nullity of T is:
iii) The Rank of T is:
3) Let T
a
b
[ d f
[def
=
1a2b+1c + 1d
[la+16+2c+2d
2a+36 + 4c+ 2d -la + (-3)6 + (-1)c+2d]
la+(-1)b+1c+ (-1)d + 2e+4f
2a+(-1)b+4c+4d + 6e + 12f
Oa+(-1)b+(-1)c+ (-4)d + (-1)e + (-2) f
6a+(-3)b+11c + 10d + 18e + 34f
i) A basis for Ker(T) would be:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F457ee6ed-b75d-44ac-a776-ab4ad4b4e512%2F53514a91-44a9-48e9-a19b-f8ee4ee286c3%2F8rpwkxq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For each linear transformation below, state a basis for Ker(T), the Nullity of the transformation, and the Rank of the transformation. If you do not
need a vector, then place zeros for all entries of that vector (for example, if you only need 2 vectors for the basis, then fill in the first
two vectors and make all subsequent vectors have 0 for all boxes).
a
1) Let T b
- B - L
la+1b+1c
2a+3b+3c]
-la + (-2)b+ (-2)c 3a+3b+3c
i) A basis for Ker(T) would be:
ii) The Nullity of T is:
iii) The Rank of T is:
2) Let T(a + bx + cx² + dx³) =
i) A basis for Ker(T) would be:
ii) The Nullity of T is:
iii) The Rank of T is:
3) Let T
a
b
[ d f
[def
=
1a2b+1c + 1d
[la+16+2c+2d
2a+36 + 4c+ 2d -la + (-3)6 + (-1)c+2d]
la+(-1)b+1c+ (-1)d + 2e+4f
2a+(-1)b+4c+4d + 6e + 12f
Oa+(-1)b+(-1)c+ (-4)d + (-1)e + (-2) f
6a+(-3)b+11c + 10d + 18e + 34f
i) A basis for Ker(T) would be:
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